This book summarizes the developments over the past several decades in the field of strong interactions at high energy. This is the first ever book almost entirely devoted to the physics of parton saturation and the color glass condensate (CGC).

Our main goal in this book is to introduce the reader systematically to the ideas, problems, and methods of high energy quantum chromodynamics (QCD). Over the years, these methods and ideas have led to a new physical picture of high energy hadronic and nuclear interactions, representing them as the interactions of a very dense system of tiny constituents (quarks and gluons) having only a small value of the QCD coupling constant. Owing to the high density of gluons and quarks the interactions in such systems are inherently nonperturbative; nevertheless, a theoretical description of these interactions is possible due to the smallness of the QCD coupling. Our main goals in the book are to show how these new ideas arise from perturbative QCD and to enable the reader to enjoy the beauty and simplicity of these emerging methods and equations.

The book's intended audience is advanced graduate students, postdoctoral fellows, and mature researchers from the neighboring subfields of nuclear and particle physics. We assume that graduate student readers are familiar with quantum field theory at the level of a standard graduate-level course based on the textbooks by Peskin and Schroeder (1995) or Sterman (1993).

We now begin the presentation of our main subject: high energy QCD, also known as small-x physics. We argue that at small Bjorken x it is natural to try to resum leading logarithms of 1/x, that is, powers of αs ln 1/x. Resummation of this parameter in the linear approximation corresponding to low parton density is accomplished by the Balitsky–Fadin–Kuraev–Lipatov (BFKL) evolution equation, which we describe in this chapter using the standard approach based on Feynman diagrams. Note that our derivation of the BFKL equation in this chapter is rather introductory in nature; a more rigorous re-derivation employing LCPT is left until for the next chapter. We point out some problems with the linear BFKL evolution; in particular we argue that it violates unitarity constraints for the scattering cross section. We describe initial attempts to solve the BFKL unitarity problem by introducing nonlinear corrections to the BFKL evolution, resulting in the Gribov–Levin–Ryskin and Mueller–Qiu (GLR–MQ) evolution equation. We discuss properties of the GLR–MQ evolution equation and, for the first time, introduce the saturation scale Qs.

Paradigm shift

Our goal in this book is to study the high energy behavior of QCD. In the context of DIS the high energy asymptotics can be explored by fixing the photon virtuality Q2 and taking the photon–proton center-of-mass energy squared ŝ to be large. In this limit the Bjorken-x variable becomes small, as follows from Eq. (2.6).

Quantum chromodynamics (QCD) is the theory of strong interactions. This is an exciting physical theory, whose Lagrangian deals with quark and gluon fields and their interactions. At the same time, quarks and gluons do not exist as free particles in nature but combine into bound states (hadrons) instead. This phenomenon, known as quark confinement, is one of the most profound puzzles of QCD. Another amazing feature of QCD is the property of asymptotic freedom: quarks and gluons tend to interact more weakly over short distances and more strongly over longer distances.

This book is dedicated to another QCD mystery: the behavior of quarks and gluons in high energy collisions. Quantum chromodynamics is omnipresent in high energy collisions of all kinds of known particles. There are vast amounts of high energy scattering data on strong interactions, which have been collected at accelerators around the world. While these data are incredibly diverse they often exhibit intriguingly universal scaling properties, which unify much of the data while puzzling both experimentalists and theorists alike. Such universality appears to imply that the underlying QCD dynamics is the same for a broad range of high energy scattering phenomena.

The main goal of this book is to provide a consistent theoretical description of high energy QCD interactions. We will show that the QCD dynamics in high energy collisions is very sophisticated and often nonlinear. At the same time much solid theoretical progress has been made on the subject over the years.

The observables discussed in this book so far have been limited to total cross sections and the related structure functions. To calculate these quantities one does not need to impose any constraints on the final state. We now present a small-x calculation of a more exclusive quantity, the cross section for diffractive dissociation, where one requires that the final state has at least one rapidity gap, i.e., a region of rapidity where no particles are produced. We again tackle the problem using the two-step formalism of Chapters 4 and 5: we first calculate the cross sections for quasi-elastic processes using the classical MV/GGM approximation and then include small-x evolution corrections in the resulting expression. For diffractive dissociation where the produced hadrons have large invariant mass, we develop a nonlinear evolution equation that describes the process.

General concepts

Diffraction in optics

Diffraction is a typical process in which we can see the wave nature of particles. When thinking of diffraction one usually pictures the diffraction of light, when a plane wave is incident on an aperture or an obstacle (see Fig. 7.1) and forms a diffraction pattern on the screen behind. The diffraction pattern consists of a bright spot in the middle and a series of maxima and minima of light intensity around it, as shown schematically in Fig. 7.1. The positions of these maxima and minima depend on the size R of the obstacle or aperture (the target), the distance d between the target and the screen (the detector), and the light wavelength λ.

The research on saturation/CGC physics is ongoing, with a number of open theoretical and phenomenological questions. Therefore, instead of conclusions, in this chapter we briefly review the phenomenology of saturation/CGC physics and list some important open theoretical problems.

Comparison with experimental data

In this section we will give a brief overview of how high energy QCD theory compares with the current experimental data. The reader may wonder whether such a comparison is possible to fit into one short section; indeed, a comparison of saturation/CGC physics with experiment could be a subject for a separate book. However, a serious quantitative comparison with experiment suffers from two major difficulties. The first is that a welldeveloped theoretical approach exists only for the scattering of a dilute parton system on a dense one; the key examples are DIS on nuclei (eA) and the proton-nucleus (pA) collisions considered earlier. At the same time, much of the data exist either for the scattering of a dense parton system on another dense parton system, as is the case in nucleus–nucleus (AA) collisions, or for the scattering of two dilute systems on each other, like DIS on a proton (ep) or proton-proton (pp) collisions. The theoretical progress in the description of these reactions in the saturation/CGC framework is rather limited, with many open questions and opportunities for further research (see Sec. 8.3 for a brief summary of the existing AA results).

Semiclassical methods based on classical solutions play an important role in quantum field theory, high energy physics, and cosmology. Real-time soliton solutions give rise both to new particles, such as magnetic monopoles, and to extended structures, such as domain walls and cosmic strings. These could have been produced as topological defects in the very early universe. Confronting the consequences of such objects with observation and experiment places important constraints on grand unification and other potential theories of high energy physics beyond the standard model. Imaginary-time Euclidean instanton solutions are responsible for important nonperturbative effects. In the context of quantum chromodynamics they resolve one puzzle—the U(1) problem—while raising another—the strong CP problem—whose resolution may entail the existence of a new species of particle, the axion. The Euclidean bounce solutions govern transitions between metastable vacuum states. They determine the rates of bubble nucleation in cosmological first-order transitions and give crucial information about the evolution of these bubbles after nucleation. These bounces become of particular interest if there is a string theory landscape with a myriad of metastable vacua.

This book is intended as a survey and overview of this field. As the title indicates, there is a dual focus. On the one hand, solitons and instantons arise as solutions to classical field equations. The study of their many varieties and their mathematical properties is a fascinating subfield of mathematical physics that is of interest in its own right. Much of the book is devoted to this aspect, explaining how the solutions are discovered, their essential properties, and the topological underpinnings of many of the solutions. However, the physical significance of these classical objects can only be fully understood when they are seen in the context of the corresponding quantum field theories. To that end, there is also a discussion of quantum effects, including those arising from the interplay of fermion fields with topologically nontrivial classical solutions, and of some of the phenomenological consequences of instantons and solitons.